Rodgers was appointed the Celtic manager in the summer of 2016 following the departure of Ronny Deila at the end of the 2015-16 season.
Celtic also progressed to the group stage of the Champions League last season, as they have done during the 2017-18 campaign.
One of the best players for the Glasgow giants over the past few years has been Scott Brown.
There have been injury problems for the 32-year-old midfielder in the past, but the Scotland international has been playing well for the Hoops this season.
Former Rangers midfielder Barton, who was on the books of the Ibrox outfit from July 2016 until November 2016, has underlined the improvement of Brown under Rodgers.
While speaking about how players at Celtic have improved under the former Liverpool boss, Barton told talkSPORT: “You look at Scott Brown, infinitely better than what he was before.”